DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. Reference herein to any social initiative (including but not limited to Diversity, Equity, and Inclusion (DEI); Community Benefits Plans (CBP); Justice 40; etc.) is made by the Author independent of any current requirement by the United States Government and does not constitute or imply endorsement, recommendation, or support by the United States Government or any agency thereof. LA-UR-25-28086 Approved for public release; distribution is unlimited. Title: Theoretical Modeling of Reactor Relevant Conditions for Plasma Jet Driven Magneto-Inertial Fusion Author(s): Cassibry, Jason; Chu, Feng Intended for: Report Issued: 2025-08-07 Los Alamos National Laboratory, an affirmative action/equal opportunity employer, is operated by Triad National Security, LLC for the National Nuclear Security Administration of U.S. Department of Energy under contract 89233218CNA000001. 1. Introduction..................................................................................................................................1 2. Theoretical Models......................................................................................................................3 2.1. Preliminary Planning for Magneto-Inertial Fusion Experiment: Confinement...................3 2.2. Power balance of an igniting sphere....................................................................................5 2.2.1. Fusion power deposition...............................................................................................5 2.2.2. Thermal conduction......................................................................................................7 2.2.3. Bremsstrahlung.............................................................................................................9 2.2.4. Mechanical Work..........................................................................................................9 2.3. Hot spot evolution and burn propagation.............................................................................9 3. Results........................................................................................................................................11 3.1. Scope of Work....................................................................................................................11 3.2. Effects of Burn Time and Local α Deposition Constraints on the Liner............................11 3.3. Fusion Power Balance as a Guide for Compressing along a Net Heating Path.................14 3.4. Point Design Burn Calculation...........................................................................................15 4. Conclusions................................................................................................................................16 5. Appendix A. Blast wave theory for estimating expansion speed of a hotspot...........................17 6. References..................................................................................................................................20 The Charger Advanced Power and Propulsion Laboratory (CAPP), a laboratory within the Propulsion Research Center (PRC) at the University of Alabama in Huntsville (UAH) is working with Los Alamos National Laboratory (LANL) to develop models and inform on promising paths for high gain magneto-inertial fusion (MIF) conditions. This report provides a framework for identifying promising conditions for achieving ignition in plasma-jet-driven magneto-inertial fusion (PJMIF)[1]. As proposed, for the first part of the contract, UAH proposes to develop a gain over unity set of stagnation conditions to provide a state of plasma conditions to achieve to set long terms goals for the PJMIF program. Specifically, UAH will model PJMIF stagnation conditions to include radiation, heat transfer, two temperature energy equations, fusion reactivity and nonlocal fusion product deposition, but no hydrodynamics for these purposes. These calculations will use a stationary plasma model to reduce simulation complexity—focusing on a DT target at 10 keV. Subsequent work will include a DD plasma layer acting as an afterburner. UAH will assume an initial magnetic field without any consideration of the topology, just assume a field strength, most likely scaled with consideration of the local hall parameter. This effort will inform the team on the tradeoff between mass, peak target field, etc and the achievable gain. PJMIF is an innovative approach to controlled fusion that bridges the gap between magnetic confinement fusion and inertial confinement fusion. This method utilizes an array of supersonic plasma jets to symmetrically compress a magnetized fuel target, achieving conditions necessary for fusion ignition. The plasma jets, typically generated by pulsed plasma accelerators, provide a scalable and cost-effective means of compression while mitigating many of the technical challenges faced by traditional approaches. PJMIF is significant because it offers a path toward high energy gain fusion, leveraging the benefits of magnetization to reduce thermal losses while utilizing kinetic compression to reach the necessary temperatures and pressures for fusion reactions. If successful, this approach could enable more compact, lower-cost fusion reactors, revolutionizing both terrestrial energy production and space propulsion applications. As global energy demands continue to rise, PJMIF stands as a promising candidate for achieving practical fusion power, bringing humanity closer to a future of abundant, clean energy. Before proceeding, this report highlights key concerns about the direction of the fusion community, which shape the framework presented. In December 2022, the National Ignition Facility (NIF) achieved a historic breakthrough in inertial confinement fusion by demonstrating fusion ignition—producing more energy from the reaction than was delivered to the fuel by the laser system[2]. This milestone validated decades of research, confirming that self-sustaining fusion burn is possible in the laboratory. NIF’s achievement represents a critical step toward practical fusion energy, reinforcing the viability of inertial confinement approaches while inspiring advancements in alternative fusion methods, including magneto-inertial fusion concepts like plasma-jet-driven compression. While the recent success of the National Ignition Facility (NIF) marks a major milestone in fusion research, traditional inertial confinement fusion (ICF) approaches face significant challenges related to size, cost, and complexity. Facilities like NIF require massive laser arrays, each demanding extreme precision and maintenance, making them prohibitively expensive and impractical for commercial power production. The intricate target fabrication and delivery systems add further engineering challenges, while the sheer scale of these facilities limits their scalability. These concerns have driven interest in alternative methods, such as plasma-jet-driven magneto-inertial fusion (PJMIF), which aims to achieve fusion conditions with a more compact, cost-effective, and potentially scalable approach. Magneto-inertial fusion has traditionally been pursued as a lower-cost, lower-volume alternative[3]. However, modern embodiments such as MagLIF require pulsed currents exceeding 50 MA of total current[4] necessitating complex pulsed power systems with pulse compression and challenging path to high pulse repetition. Alternative: Currently, PJMIF requires hundreds of pulsed plasma accelerators arranged spherically. This design process begins with anticipated stagnation conditions and confinement time, working backward to define liner requirements. Then knowing the mass achievable per plasma accelerator and standoff, determining the final system. It’s a system of the order of 100 MJ. This will be quite large, complicated, and expensive to build and maintain. Additionally, if NIF is any indicator, PJMIF will struggle for many years before maturing the technologies needed for the high precision needed for shock heating and spherical compression. While the CAPP laboratory is fully capable of modeling this pathway, this report proposes an alternative PJMIF parameter space aimed at simplifying and reducing the overall reactor size. The rest of the report is organized as follows. Section 2 presents the theoretical models used in this report, specifically preliminary planning, power balance for stationary plasma states, and analytical states. Results are presented in section 3, followed by discussion and concluding remarks in section 4. This section draws upon numerous analytical models of magneto-inertial fusion (MIF). An excellent reference for the inertial physics is the textbook by Atzeni and Meyer-ter-Vehn[5]. Among the earliest MIF studies, Lindemuth and Kirkpatrick investigated the impact of magnetic fields on inertial targets, systematically varying density and implosion velocity[6]. Their work was generalized by Kirkpatrick to power balance diagrams as a function of aerial density and fuel temperature [7]. Drake et al demonstrated that targets with an initial density of 10^24 m^-3, magnetic field of ~10 T, and initial temperature of 100 eV could reach ignition and gain over unity with ~100 kJ of initial energy. Similar conclusions about the merits of magnetic fields and preheat were identified by Ribe and Barnes[8] with magnetized impact fusion, Turchi[9] with the Linus concept, and Siemon et al [10] with magnetized target fusion. Deposition of fusion products into magnetized targets in the context of MIF was performed by Basko et al[11]. A critical parameter is the ratio of the target radius to the born-on Larmor radius of the charged fusion products. Other transport properties are documented in the NRL Plasma Formulary[12], with updates on many parameters provided by Davies et al[13]. For plasma liner specific studies, semi-analytic 1D treatments of Mag LIF[14] and Staged Z-pinch[15] were developed and performed. The focus on those studies was the verification of the tools themselves without a thorough exploration of the parameter space. Most recently, Langendorf and Hsu developed a semi-analytic model of PJMIF[16], highlighting the 1D result that gains of 3 to 30 may be possible with initial plasma liner energies of 20-40 MJ. 2.1. Preliminary Planning for Magneto-Inertial Fusion Experiment: Confinement Gain over unity in ICF and MIF plasmas are informed by the condition that the confinement time needs to exceed the fusion burn time necessary to recover the energy investment, namely the inequality \tau_{conf} \ge \tau_{fus} (1) The fusion burn time comes from a time integration of the fusion reactivity equation. In general, the volumetric fusion power given by (dE/dt)_{fus} = (n_i n_j / (1 + \delta_{ij})) \langle\sigma v\rangle_{ij} \mathcal{V} E_{fus, ij} (2) where \delta_{ij} is 1 if i=j (i.e. D-D, T-T reactions) and 0 otherwise and \mathcal{V} is the target volume. The number density of the fuel with n_i = X_i n (3) where \sum_{i=1}^{N_{species}} X_i = 1 are the mole fractions. Typically, the number of species is 2, and deuterium and tritium are the species in proportions of 50% each by mole fraction. For binary reactions involving different species, we have (dE/dt)_{fus} = n^2 X_1 (1 - X_1) \langle\sigma v\rangle \mathcal{V} E_{fus} (4) For thermal DT plasmas, neglecting the magnetic field energy, the total energy is E_{th} = n m_i \mathcal{V} (\bar{R}/(\gamma-1)) 2T. The total energy released from fusion can exceed the thermal energy by almost a factor of 500, so the number density can be approximated as a constant for the sake of rapid order of magnitude analysis, setting up an inequality E_{th} = n m_i \mathcal{V} (\bar{R}/(\gamma-1)) 2T \le E_b \approx n^2 X_1 (1 - X_1) \langle\sigma v\rangle \mathcal{V} E_{fus} \tau_b (5) This provides our definition of burn time. So for planning purposes, the time to recover the thermal energy must satisfy \tau_b \ge ( (2 m_a \bar{R} / (\gamma - 1)) (1 / (X_1 (1 - X_1))) ) ( T / (\langle\sigma v\rangle E_{fus}) ) (1 / n) (6) For practical purposes, burn time is inversely proportional to number density, and the temperature, reactivity, and fusion energy per reaction set the scale. In ICF and MIF, the confinement time is required to exceed the burn time. Confinement time from analytical arguments has historically been a point of contention and should be a focus of future experimental and modeling studies (see e.g. Thio et al [17] vs. Parks [18]). For this model it is assumed that the confinement time is the liner thickness l_L divided by the expansion rate v_{exp, L}, and it is assumed that the expansion rate is based on a fraction of the incoming liner speed, due to the rapid radiative cooling by the high Z materials, leading to liner temperatures of 2-4 eV. Further, we require dynamic pressure to exceed the stagnation and magnetic field pressure of the target, leading to an implosion speed of V_L = \sqrt{ (4 n k T_T + 0.5 B_T^2 / \mu_0) / \rho_L } (7) where \rho_L is the mass density of the liner. Assuming a liner expansion rate of V_L / M_L leads to the required thickness of the liner, l_L \ge (V_L / M_L) (2 m_a \bar{R} / (\gamma - 1)) (1 / (X_1 (1 - X_1))) (T_T / (\langle\sigma v\rangle n_T E_{fus})) (8) The magnetic field is not arbitrary, and can be guided by the ratio of the target radius to the Larmor radius of the born-on alpha particles. Using results from Ref. [11], r_{\alpha, Lar, 0} \le r_T / 3 (9) where r_T is the target radius. This inequality leads to B_T \ge \sqrt{2 m_\alpha E_{fus,\alpha 0}} / (2 q r_{\alpha, Lar, 0}) (10) With these parameters defined by the independent variable of n and r_T, everything, including the total mass, energy, and kinetic energy of the liner, can be estimated providing a promising and much narrower selection for initial parameters in an otherwise overwhelmingly large multidimensional parameter space. For the burning target, the first law of thermodynamics gives dE/dt = W_{dep} - W_m - W_r - W_e (11) where W_{dep}, W_m, W_r, and W_e are the powers of fusion product deposition, mechanical work (pdV), radiation, and thermal conduction. We treat each of these terms below. 2.2.1. Fusion power deposition In general, the volumetric fusion power given by dE/dt \equiv W_{fus} = (n_i n_j / (1 + \delta_{ij})) \langle\sigma v\rangle E_{fus} \mathcal{V} (12) where \delta_{ij} is 1 if i=j (i.e. D-D, T-T reactions) and 0 otherwise. In general, the deposition power is given by W_{dep} = W_{fus} f_{dep} = (n_i n_j / (1 + \delta_{ij})) \langle\sigma v\rangle \mathcal{V} \sum_m E_{fus, m} f_m (13) For example, the power density for equimolar DT has already been given as (if n/2 = n_D = n_T, also noting \delta_{ij} = 0), dE_{DT}/dt \equiv W_{fus} = (1/4) n^2 \langle\sigma v\rangle_{DT} E_{DT} (14) A fraction of this power f_{dep} is deposited within the hotspot. We can write this as W_{dep} = W_{fus} f_{dep} = W_\alpha (f_\alpha + 4 f_n) (15) for DT specifically. For temperatures below 25 to 30 keV, charged particle fusion products slow down mostly due to small angle electron scattering collisions. The velocity decreases according to dv_\alpha/dt = - v_\alpha / (2 t_{\alpha e}) (16) where t_{\alpha e} is a characteristic time for energy deposition given by t_{\alpha e} \approx 42 T_e^{3/2} / (\rho \ln\Lambda_{\alpha e}) [ps] (17) where T_e is in units of keV and \ln\Lambda_{\alpha e} is the Coulomb logarithm for collisions between alphas and electrons. The range of a 3.5 MeV alpha particle in a homogeneous plasma is obtained as l_\alpha = 2 v_{\alpha 0} t_{\alpha e} \approx 0.107 T_e^{3/2} / (\rho \ln\Lambda_{\alpha e}) [cm] (18) where v_{\alpha 0} = 1.29 \times 10^9 cm/s is the 'born on' velocity of the alpha particles. More conveniently, the approximate formula for l_\alpha is l_\alpha = 1.07 T_e^{3/2} / (\rho \ln\Lambda_{\alpha e}) m (19) where \rho is the mass density of the fuel in kg/m^3, T_e is the temperature in keV, and \ln\Lambda_{\alpha e} is the Coulomb logarithm (impact parameter). Typically about 10.0 regardless of the physics regime. The fraction of the energy deposition by the alpha is f_\alpha = (3/2)\tau_\alpha - (4/5)\tau_\alpha^2 for \tau_\alpha \le 1/2; f_\alpha = 1 - 1/(4\tau_\alpha) + 1/(160\tau_\alpha^3) for \tau_\alpha \ge 1/2 (20) where \tau_\alpha = R_h / l_\alpha is the ratio of the burning sphere to 'range of alpha particle'. A magnetic field will enhance the deposition. The most important figure of merit is the ratio of the radius Rh with the Larmor radius of an alpha, which is r_{L\alpha} = m v_{\perp\alpha 0} / (2 q B) (21) where v_{\perp\alpha 0} is the 'born on' velocity: E_\alpha = 0.5 m_\alpha v_{\perp\alpha 0}^2 (22), thus v_{\perp\alpha 0} = 1.2986 \times 10^7 m/s. \tau_\alpha = (R_h / l_\alpha) (1 + R_h / r_{L\alpha}) \equiv \bar{R}(1 + b) (23) Figure 1 shows fractional fusion deposition as a function of target radius to stopping length at fixed value of target radius to born on alpha Larmor radius (b = 0.1, 1, 10). Electrons dominate thermal conduction because they are highly mobile. There are three thermal conductivities for electrons, k_{||,e}, k_{\perp,e}, k_{\wedge,e}. For the perpendicular component, k_{\perp,e} = (n T_e \tau_e k_B^2 / m_e) ( (f_{\gamma 1}' x^2 + f_{\gamma 0}') / \Delta ) [W / (m \cdot K)] (24) where temperature is in K, Z is the charge (1.0 for fully ionized H, D, and T). The electron collision time \tau_e is given by \tau_e = (3 \sqrt{m_e} / (4 \sqrt{2\pi})) ( (k_B T_e)^{3/2} / (n_e \lambda) ) ( (4\pi \epsilon_0)^2 / q^4 ) (25) ne is electron number density in m^-3, me is electron mass in kg. x is the Hall parameter: x = \omega_e \tau_e = (q B / m_e) \tau_e (26) \Delta = x^4 + f_{\delta 1}' x^2 + f_{\delta 0}' (27) Braginskii coefficients: f_{\gamma 0}' = 11.92( 0.10067 + 0.58456 / (-0.35 + Z^{1.1}) ) f_{\gamma 1}' = 4.664( 0.69683 + 0.30317 / Z ) f_{\delta 0}' = 3.7703( 0.025489 + 0.63343 / (-0.35 + Z^{3/2}) ) f_{\delta 1}' = 14.79( 0.50588 + 0.40765 / (-0.175 + Z) ) (28) Coulomb logarithm \lambda: \lambda = 29.9 - \log(\sqrt{n_e} T_{eV}^{-3/2}) for T_{eV} < 10 eV; \lambda = 30.9 - \log(\sqrt{n_e} T_{eV}^{-1}) for T_{eV} \ge 10 eV (29) For ions: k_{\perp,i} = (n T_i \tau_i k_B^2 / (m_a MW_i)) ( (2 x_i^2 + 2.645) / \Delta_i ) [W / (m \cdot K)] (30) \tau_i = (3 \sqrt{m_e} / (4 \sqrt{\pi})) ( (k_B T_i)^{3/2} / (n_i \lambda) ) ( (4\pi \epsilon_0)^2 / q^4 ) (31) x_i = \omega_i \tau_i = (q B / (m_a MW_i)) \tau_i (32) \Delta_i = x_i^4 + 4.03 x_i^2 + 2.33 (33) Thermal conduction power on average: W_e = -k_e \nabla T_e S (34) \nabla T_e \approx T_e / r_T, yielding: W_e = - (2.91 \times 10^4 T_{e,eV}^{7/2} / (Z \ln\Lambda R_h^2)) ( (f_{\gamma 1}' x^2 + f_{\gamma 0}') / \Delta ) \mathcal{V} (35) 2.2.3. Bremsstrahlung Bremsstrahlung power per unit volume is [20]: P_{br} = \mathcal{V} 4.856 \times 10^{-37} n_i^2 T_{e,keV}^{1/2} Z_i^3 (36) 2.2.4. Mechanical Work W_m = p_h (dV/dt) (37) For a sphere, dV/dt = 4\pi R_h^2 (dR_h/dt) = 4\pi R_h^2 u (38) Thus W_m = 4\pi r_T^2 u \rho_T R_T T_T (39) For isobaric ignition, u = 0. 2.3. Hot spot evolution and burn propagation Energy conservation in the burning region: M (de/dt) = (W_{cp} f - W_b - W_e) \mathcal{V} - p S u_{exp} - e (dM/dt) (40) Surface area S = 4\pi r^2 (spherical) or 2\pi r l (cylindrical) (41) Rate of mass accretion: e (dM/dt) = [W_{cp}(1 - f_{cp}) + W_e] \mathcal{V} (42) Total energy E = e M (43) Specific internal energy e = C_v T = (\bar{R} / (MW (\gamma - 1))) (T_e + T_i) (44) Volume \mathcal{V} = (4/3)\pi r^3 (spherical) or \pi r^2 l (cylindrical) (45) Burn wave velocity: dr/dt \equiv u_{bw} = (1 / (4\pi\rho)) ( (M/\rho) (3 / (4\pi)) )^{-2/3} (dM/dt) (spherical) or (1 / (2\pi\rho l)) ( (M/\rho) (1 / (l\pi)) )^{-1/2} (dM/dt) (cylindrical) (47) Pressure is approximated with ideal gas: P = n_i k T_i + n_e k T_e (48) 3.1. Scope of Work The modeling was performed to guide multidimensional modeling using SPFMax. Burn time, magnetic field, plasma liner velocity, liner thickness, total mass, and total kinetic energy were evaluated. Power balance diagrams guide compression along net heating paths. 3.2. Effects of Burn Time and Local α Deposition Constraints on the Liner Burn time is inversely proportional to density (varying from ~10 μs down to 20 ns for densities 10^25 to 10^28 m^-3). If the afterburner ignites via heat transfer and fusion product transfer from the target, the target's role shifts from high yield to spark ignition, reducing required liner kinetic energy by an order of magnitude. Minimum magnetic field and plasma beta (\beta) are plotted in Fig. 3. Target radii of 10 mm and densities of 10^26 to 10^27 m^-3 place \beta between 10 and 100. Railguns are recommended for liner formation (1 g per gun) due to handling higher mass per shot, mitigating complexity of hundreds of coaxial plasma guns. 3.3. Fusion Power Balance as a Guide for Compressing along a Net Heating Path Power balance contour plots (Fig. 8) demonstrate that cylindrical compression with modest compression rates (tens of km/s) and B = 50 T can reach breakeven. Thermal energy scales with r_T^3 (Fig. 9): a 10 keV 1 cm target at 10^26 m^-3 contains 2 MJ of thermal energy, while a 0.5 cm target contains only 250 kJ. Reducing final target radius by a factor of 2 lowers capacitor bank requirements from 20 MJ to 2.5 MJ. 3.4. Point Design Burn Calculation A point design calculation for a target with 0.5 cm initial radius, 50 T magnetic field, and 10^26 m^-3 ion density demonstrates unity gain reached on the order of 1 μs (Fig. 10). 1. Increased liner density substantially reduces kinetic energy required for compression, enabling fusion-relevant conditions at sub-megajoule scales. 2. Optimal target parameters: ion density of 10^24 to 10^25 m^-3, initial radius ~10 cm, compression velocity ~50,000 m/s. 3. Constraining magnetic field so born-on Larmor radius is smaller than target radius (50-100 T at stagnation), keeping plasma \beta well above unity to avoid MHD instabilities. 4. Coupling between liner pressure and target internal pressure is critical for maintaining compression and burn propagation. 5. Railguns recommended for liner formation, with short-term focus on medical isotope production, materials testing, and space propulsion. Gain exceeding unity is achievable with a 10 MJ system or smaller by utilizing spark ignition of a cold fuel layer. In blast wave theory [23], the general form for shock radius is: R_s(t) = (E t^2 / (\alpha A))^{1 / (\nu + 2 - \omega)} (49) \dot{R}_s(t) = (2 R_s(t)) / ( (\nu + 2 - \omega) t ) (50) Undisturbed density: \rho_0(r) = A r^{-\omega} (51) Dimensionless variables: \xi = r / R_s(t) (52) D(\xi) = \rho(r,t) / \rho_s (53) V(\xi) = v(r,t) / v_s (54) P(\xi) = p(r,t) / p_s (55) Self-similar blast wave coefficients and algebraic solutions: \omega_1 = (3\nu - 2 + \gamma(2-\nu)) / (\gamma + 1) (56) \omega_2 = (2(\gamma - 1) + \nu) / \gamma (57) \omega_3 = \nu(2 - \gamma) (58) \beta_0 = 1 / (\nu\gamma - \nu + 2) (59) \beta_6 = 2 / (\nu + 2 - \omega) (60) \beta_7 = \omega \beta_6 (61) \beta_8 = \nu \beta_6 (62) \beta_2 = (\gamma - 1) / (\gamma (\omega_2 - \omega)) (63) \beta_3 = (\nu - \omega) / (\gamma (\omega_2 - \omega)) (64) \beta_1 = \beta_2 + (\gamma + 1)\beta_0 - \beta_6 (65) \beta_4 = \beta_1 ((\nu - \omega)(\nu + 2 - \omega)) / (\omega_3 - \omega) (66) \beta_5 = (2\nu - \omega(\gamma + 1)) / (\omega_3 - \omega) (67) C_0 = 2^\nu \pi^{(\nu-1)/2} \Gamma((\nu+1)/2) / \Gamma(\nu) (68) C_5 = 2 / (\gamma - 1) (69) C_6 = (\gamma + 1) / 2 (70) C_4 = (\nu + 2 - \omega) \beta_0 C_6 (71) C_1 = \gamma C_5 (72) C_2 = C_6 / \gamma (73) C_3 = (\nu\gamma - \nu + 2) / ((\omega_1 - \omega) C_6) (74) Dimensionless profile equations across parameter F from C_2 to 1: \xi = F^{-\beta_6} [C_1(F - C_2)]^{\beta_2} [C_3(C_4 - F)]^{-\beta_1} (77) D = F^{\beta_7} [C_1(F - C_2)]^{\beta_3 - \omega\beta_2} [C_3(C_4 - F)]^{\beta_4 + \omega\beta_1} [C_5(C_6 - F)]^{-\beta_5} (78) V = \xi F (79) P = F^{\beta_8} [C_3(C_4 - F)]^{\beta_4 + (\omega-2)\beta_1} [C_5(C_6 - F)]^{1 - \beta_5} (80) \alpha = (8 C_0 / ((\gamma^2 - 1)(\nu + 2 - \omega)^2)) \int_0^1 \xi^{\nu-1} (D V^2 + P) d\xi (81) Shock front jump conditions: \rho_s = ((\gamma + 1)/(\gamma - 1)) \rho_0 (82) v_s(t) = 2 \dot{R}_s(t) / (\gamma + 1) (83) p_s(t) = 2 \rho_0 \dot{R}_s(t)^2 / (\gamma + 1) (84) \rho(r,t) = \rho_s D(\xi) (85) v(r,t) = v_s(t) V(\xi) (86) [1] A. 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